Dear Editor, dear Referee,

we revised once more our manuscript taking into account the Referee's comments, suggestions, and criticisms.  We hope that the improved manuscript is now suitable for publication.

Below you will find a detailed, point-by-point, response, where all changes to the text are also outlined.

Referee: Regarding the first main point I asked you in my previous report: your discussion of how the variance in RM can be used to determine or constrain the variance in scattering is very thorough. What I am missing is the contribution of the large-scale field perpendicular to the line of sight, which in light of your toy model from section 3.3, moves the centre of cloud of realisations in a coherent direction. If you have investigated this with your model of the Milky Way (described in Pshirkov et al. 2011) then I would ask you to write this, which would make your statements at the end of section 3.2 stronger. Now you only write that you checked that scattering angles are smaller than the bins you are using, which can be made more specific to address my current question. 

Reply: We have mentioned our reference model Pshirkov et al. 2011 at the end of section 3.2, in such a way that this issue is not only presented but also practically settled.  The effects of both the regular (checked a priori) and turbulent (checked a posteriori) magnetic fields are thus under control.

Referee: When I studied your Appendix on two consecutive days I realised that on the second day I had started interpreting <RM^2> as the average of the squares of the rotation measures, while you in fact are using RM from the middle of section 3.2 as sqrt(var(RM)), which you also referred to as 'sigma' in section 3.1. Similarly, you have used 'delta' to mean different things throughout the manuscript. While the meaning can generally be derived from the context, sometimes the reader get be put on the wrong footing. I would ask you to make your use of the different symbols more clear, especially when you introduce a variable early in the manuscript and then assign a new meaning to this variable later on. 

Reply: We have clearly specified that the identification of \sigma for \re RM^2 \ra is valid only within that section, to not clutter notation too much.  Then, the deflection is now called \vartheta and the displacement is \Theta.  We believe that all symbols are now unambiguous.

Referee: Regarding the scale height of the free electron density, the reason you give for using the thick disk from Gaensler et al. (2008) leads to internal inconsistencies, which I tried to explain in my previous referee report. I went back to your paper from 2011, where you discuss electron density models in section 3.3. There you explain that a more extended thick disk describes the RM data that you analysed better, but I understand that you only considered the original version of NE2001 and a new version of NE2001 with a thick disk of 1.8 kpc. Therefore I would be curious to see what happens if you were to combine NE2001 with the thick disk of 1.3 kpc I derived (or use TC93 with a scale height of 1.6 kpc); maybe this works even better than combining NE2001 with a thick disk of 1.8 kpc. I agree that by being explicit in your assumptions the reader can rescale <n> from Eqn. (8), but I would consider 1.3 kpc to be the right reference value instead of 1.8 kpc (in combination with NE2001), based on my explanation from the first referee rapport. 

Reply: We have run the code with the 1.3 thickness for the NE2001 model; we observe that qualitatively the picture is the same, and quantitavely there is only a 20% shift in the deflections.  We have reported about this run in the second half of section 3.4, where we discuss all modifications to our baseline parameters.

Referee: Equation 7: if possible, please add a few lines to the appendix where you derive this equation. 

Reply: Actually, this is already explained in the appendix, starting before equation (17) and below.

Referee: p3 column 1 line 51: the third term in equation 6 also depends on <B>. 

Reply: Yes, that is correct; however, this formula wants to emphasise the role of the electron density fluctuations, which are poorly known.  While we mention all possible terms here, we do not go into the details in the main text; those are to be found in the appendix.

Referee: p4 column 2 line 14: this is a very long sentence, and it loses clarity towards the end. Please rephrase. 

Reply: We have broken down the sentence in three, which should be a lot more readable.

Referee: p4 column 2 line 31: so far you have only considered the scale height of free electrons, but you should also discuss the scale height of the large-scale and small-scale field components. 

Reply: Our derivation of upper limits is based on the assumption that regions of considerable magnetic field and electron density are largely spatially coincident as discussed in Section 4.  On the other hand if the magnetic fields are more strongly concentrated it could only strengthen our upper limits obtained with Eq (8). 

Most of the models for the regular disk field predicts its vertical scale smaller than the scale of WHIM (less than 1 kpc) thus justifying our approach.  There is much less agreement on the vertical scale of the turbulent component but we think that it could be safely assumed that it shouldn't be significantly larger than the vertical scale of the electron density.

We have spelt out once more also in section 3.4 what our assumption is in this respect.

Referee: p4 column 2 line 57: what do you base this value of 30% on? When you integrate an exponential thick disk with midplane density n_0 and scale height H out to infinity, the DM_infty = n_0 * H/sin|b|, which is exactly the same as the n_0 * D (where D = your d/sin|b|) you derive.

Reply: This number is a rough extimate which comes from either numerical integration or even more simply by integrating a step function versus the (normalised) exponential or sinh.  We have clarified this in the text.


Sincerely,

Maxim Pshirkov, Peter Tinyakov, Federico Urban
